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Re: Ye Olde Arte has a large supply of colored poster board. The colors ar [#permalink]
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If we have only 3 colors for 5 boards, there are only 2 possible options 3 (first color) + 1 (second color) + 1 (third color) and 2(first color) + 2 (second color) + 1 (third color). There is 3 different ways for each option.

For example, option 1 if we have red, green, blue:

red + green + blue + blue + blue
red + green + green + green + blue
red + red + red + green + blue

option 2:

red + red + green + green + blue
red + red + green + blue + blue
red + green + green + blue + blue

So, \(C^3_1\) actually means that we choose one color out of 3 for 3 boards (option 1) or for 1 board (option 2)
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Re: Ye Olde Arte has a large supply of colored poster board. The colors ar [#permalink]
For 3 colors: one color of 3 for 3+1+1 combination and one color of 3 for 2+2+1 combination.

Can we write for the option 2 as 6C3 * 3C2 (which is equal to 6C3 * 3C1) ?

This is because let us say we have chosen 3 colors for 3 boards, and now we need to choose 2 colors out of 3 for the remaining 2 boards.

So option is 6C3 * 3C1 + 6C3 * 3C2.
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Re: Ye Olde Arte has a large supply of colored poster board. The colors ar [#permalink]
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Yup, \(C^3_2\) is correct too.
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Re: Ye Olde Arte has a large supply of colored poster board. The colors ar [#permalink]
good question indeed.
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Re: Ye Olde Arte has a large supply of colored poster board. The colors ar [#permalink]
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Re: Ye Olde Arte has a large supply of colored poster board. The colors ar [#permalink]
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